Here’s a pretty idea. A Markov chain is one of the simplest forms of dependence in random variables: an infinite sequence of dependent random variables, where the probability distribution of the next random variable only depends on the value of the current random variable. If you reverse the sequence of variables, you get another Markov chain, the reverse Markov chain. Some Markov chains, reversible Markov chains, have the property that when you reverse them, you get back the same chain. Markov chains represent processes that have no history, in that future is determined solely by the present, not the past. A reversible Markov chain not only has no history, but time has no direction.

Here is a draft of a book by Aldous and Fill on the theory of reversible Markov chains.

There is a notion in symbolic dynamics of a “topological Markov chain” that is analogous to a Markov chain in probability theory. It’s occurred to me that you can extend the analogy to a complete analogy with probability theory. We’re still interested in sets of events, but now we’re no longer interested in the probability of an event, but just whether or not an event is possible.

Start with a σ-algebra of sets, as usual. Instead of associating a probability with each set, associate a member of the set {Not Possible, Possible}. The empty set is assigned the value Not Possible, while the whole space is assigned the value Possible. A countable disjoint union of sets is Possible if and only if at least one of the individual sets is Possible.

A measure takes values in the semigroup of the nonnegative real numbers closed under addition. Here, we’ve replaced that semigroup with the semigroup of {Not Possible, Possible} under the commutative binary operation +, with multiplication table:

I’ll explain the relationship with topological Markov chains in a future post.